Dyadic Deontic Logic
Axiom 5(a) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that \(\emptyset \notin \operatorname{ob}(X)\) for all \(X\in \mathcal P(U)\).
Axiom 5(b) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that
\[ ∀ (X Y Z : Set U), Z ∩ X = Y ∩ X → (Z ∈ ob X ↔ Y ∈ ob X) \]
Axiom 5(d) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that
\[ ∀ (X Y Z : Set U), Y ⊆ X → Y ∈ ob X → X ⊆ Z → Z \ X ∪ Y ∈ ob Z \]
Axiom 5(bd) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that \(\forall (X Y Z \in \mathcal PU), Y \in \operatorname{ob}X \land X \subseteq Z \to Z \setminus X \cup Y \in \operatorname{ob}Z\).
If \(\operatorname{ob}\) satisfies 5(b) and 5(d) then it satisfies 5(bd).